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1. Operator learning for homogenizing hyperelastic materials, without PDE data

   https://doi.org/10.1016/j.mechrescom.2024.104281  

   Zhang, Hao; Guilleminot, Johann (June 2024, Mechanics Research Communications)

   In this work, we address operator learning for stochastic homogenization in nonlinear elasticity. A Fourier neural operator is employed to learn the map between the input field describing the material at fine scale and the deformation map. We propose a variationally-consistent loss function that does not involve solution field data. The methodology is tested on materials described either by piecewise constant fields at microscale, or by random fields at mesoscale. High prediction accuracy is obtained for both the solution field and the homogenized response. We show, in particular, that the accuracy achieved with the proposed strategy is comparable to that obtained with the conventional data-driven training method. 

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2. HOMOGENIZATION OF QUASI-CRYSTALLINE FUNCTIONALS VIA TWO-SCALE-CUT-AND-PROJECT CONVERGENCE

   https://doi.org/10.1137/20M1341222  

   Ferreira, R.; Fonseca, I.; Venkatrama, R.n (January 2021, SIAM journal on mathematical analysis)

   We consider a homogenization problem associated with quasi-crystalline multiple inte- grals of the form uε ∈ Lp(Ω;Rd) 7→ ˆΩ fR x, xε,uε(x) dx, where uε is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, fR, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields uε that are in the kernel of a given constant-coefficient linear partial differential operator, A, that is, Auε = 0. Our results provide a generalization of related ones in the literature concerning the A = curl case to more general differential operators A with constant coefficients, and without coercivity assumptions on the Lagrangian fR. 

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3. Dual Neural Network (DuNN) method for elliptic partial differential equations and systems

   https://doi.org/10.1016/j.cam.2025.116596  

   Liu, Min; Cai, Zhiqiang; Ramani, Karthik (March 2025, Journal of Computational and Applied Mathematics)

   This paper presents the Dual Neural Network (DuNN) method, a physics-driven numerical method designed to solve elliptic partial differential equations and systems using deep neural network functions and a dual formulation. The underlying elliptic problem is formulated as an optimization of the complementary energy functional in terms of the dual variable, where the Dirichlet boundary condition is weakly enforced in the formulation. To accurately evaluate the complementary energy functional, we employ a novel discrete divergence operator. This discrete operator preserves the underlying physics and naturally enforces the Neumann boundary condition without penalization. For problems without reaction term, we propose an outer-inner iterative procedure that gradually enforces the equilibrium equation through a pseudo-time approach. 

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4. Source term method for binary neutron stars initial data

   https://doi.org/10.1088/1361-6382/abfc29  

   Tsao, Bing-Jyun; Haas, Roland; Tsokaros, Antonios (June 2021, Classical and Quantum Gravity)

   Abstract The initial condition problem for a binary neutron star system requires a Poisson equation solver for the velocity potential with a Neumann-like boundary condition on the surface of the star. Difficulties that arise in this boundary value problem are: (a) the boundary is not known a priori , but constitutes part of the solution of the problem; (b) various terms become singular at the boundary. In this work, we present a new method to solve the fluid Poisson equation for irrotational/spinning binary neutron stars. The advantage of the new method is that it does not require complex fluid surface fitted coordinates and it can be implemented in a Cartesian grid, which is a standard choice in numerical relativity calculations. This is accomplished by employing the source term method proposed by Towers, where the boundary condition is treated as a jump condition and is incorporated as additional source terms in the Poisson equation, which is then solved iteratively. The issue of singular terms caused by vanishing density on the surface is resolved with an additional separation that shifts the computation boundary to the interior of the star. We present two-dimensional tests to show the convergence of the source term method, and we further apply this solver to a realistic three-dimensional binary neutron star problem. By comparing our solution with the one coming from the initial data solver cocal, we demonstrate agreement to approximately 1%. Our method can be used in other problems with non-smooth solutions like in magnetized neutron stars. 

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5. Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

   https://doi.org/10.1007/978-3-319-97136-0_3  

   Wu, Yangqingxiang; Zikatanov, Ludmil T (January 2018, Lecture notes in computer science)

   Sistek, J.; Tichy, P; Kozubek, T.; Cermak, M.; Lukas, D.; Jaros, J.; Blaheta, R. (Ed.)

   We introduce an efficient method for computing the Stekloff eigenvalues associated with the indefinite Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the discretized problem with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions operators. The proposed Fourier method, combined with proper eigensolver, results in an efficient and easy approach for computing the Stekloff eigenvalues 

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